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Thermal Properties of Matter

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Question : 6 of 22
Marks: +1, -0
A steel tape 1 m long is correctly calibrated for a temperature of 27.0°C. The length of a steel rod measured by this tape is found to be 63.0 cm on a hot day when the temperature is 45.0°C. What is the actual length of the steel rod on that day? What is the length of the same steel rod on a day when the temperature is 27.0°C? Coefficient of linear expansion of steel =1.20×10−5K−1.= 1.20 \times 10^{-5} K^{-1}.
Solution:  
Here, length of the steel tape at 27°C is 100 cm.
∴L=100 cm,T=27∘C\therefore L = 100 \text{ cm}, T = 27^{\circ}\text{C}
The length of steel tape at 45°C is L′=L+ΔLL' = L + \Delta L
L′=L+αLΔL(∵ΔL=αLΔL)L' = L + \alpha L \Delta L (\because \Delta L = \alpha L \Delta L)
L′=L+αL(T2−T1)L' = L + \alpha L (T_2 - T_1)
L′=100+(1.20×10−5)×100×(45∘−27∘)=100.0216 cmL' = 100 + (1.20 \times 10^{-5}) \times 100 \times (45^{\circ} - 27^{\circ}) = 100.0216 \text{ cm}
Length of 1 cm mark at 27°C on this scale, at 45°C =100.0216100= \frac{100.0216}{100}
Length of 63 cm measured by this tape at 45°C will be =100.0216100×63=63.0136 cm= \frac{100.0216}{100} \times 63 = 63.0136 \text{ cm}
When temperature is 27°C, the size of 1 cm mark on the steel tape will be exactly 1 cm as the steel tape has been calibrated at 27°C.
Therefore, length of the steel of the rod at 27∘C=63×1=63 cm27^{\circ}\text{C} = 63 \times 1 = 63 \text{ cm}
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